Cobordisms of sutured manifolds and the functoriality of link Floer homology

Abstract: It has been a central open problem in Heegaard Floer theorywhether cobordisms of links induce homomorphisms on the associated linkFloer homology groups. We provide an affirmative answer by introducing anatural notion of cobordism between sutured manifolds, and showing that sucha cobordism induces a map on sutured Floer homology. This map is a commongeneralization of the hat version of the closed 3-manifold cobordism map inHeegaard Floer theory, and the contact gluing map defined by Honda, Kazez,and Mati´c. We show that sutured Floer homology, together with the abovecobordism maps, forms a type of TQFT in the sense of Atiyah. Applied to thesutured manifold cobordism complementary to a decorated link cobordism,our theory gives rise to the desired map on link Floer homology. Hence, linkFloer homology is a categorification of the multi-variable Alexander polynomial.We outline an alternative definition of the contact gluing map usingonly the contact element and handle maps. Finally, we show that a Weinsteinsutured manifold cobordism preserves the contact element.